<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Proof</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;y_1~ \textrm{is a solution}:\quad y_1^{\prime \prime}+p(x) y_1^{\prime}+q(x) y_1=0,\\
&amp;y_2~ \textrm{is a solution}:\quad y_2^{\prime \prime}+p(x) y_2^{\prime}+q(x) y_2=0.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Let <span class="process-math">\(y=C_1 y_1+C_2 y_2\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
LHS&amp;=[C_1 y_1+C_2 y_2]^{\prime \prime}+p(x) [C_1 y_1+C_2 y_2]^{\prime}+q(x) [C_1 y_1+C_2 y_2]\\
&amp;=C_1 [y_1^{\prime \prime}+p(x) y_1^{\prime}+q(x) y_1]+C_2 [y_2^{\prime \prime}+p(x) y_2^{\prime}+q(x) y_2]\\
&amp;=0=RHS.
\end{aligned}
\end{equation*}
</div>
<span class="incontext"><a href="sec3_2.html#p-75" class="internal">in-context</a></span>
</body>
</html>
